Theorems · Theorem · category theory

CategoryTheory.Presheaf.nonempty_isLimit_mapCone_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} J]
  [inst_2 : CategoryTheory.LocallySmall.{w, v, u} C] {F : CategoryTheory.Functor J Cᵒᵖ}
  (c : CategoryTheory.Limits.Cone F)
  {c' : CategoryTheory.Limits.Cocone (F.leftOp.comp CategoryTheory.shrinkYoneda.{w, v, u})}
  (hc' : CategoryTheory.Limits.IsColimit c') (P : CategoryTheory.Functor Cᵒᵖ (Type w)),
  Nonempty (CategoryTheory.Limits.IsLimit (P.mapCone c)) ↔
    (CategoryTheory.MorphismProperty.single (CategoryTheory.Presheaf.coconePtToShrinkYoneda c hc')).isLocal P

Defined in

Mathlib.CategoryTheory.Limits.Types.PreservesLimit

Cited by

1 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.LocallySmall